Having a domain name is separate from having a web host. There is an organization named the "Internet Corporation for Assigned Names and Numbers" aka ICANN. They subcontract most of their work to subcontractors who sell domain names to the public. Within very broad limits, you can make up any name you like and, if it isn't already taken, it is yours. Depending on who your registrar is, the cost for registering a domain name is $10 per year or less. At this rate it can make sense to register a name even if you have no immediate use for it just to hold it for later use.
Usually, you will register your domain name from your hosting company. There are companies that are domain name registrars only and they often offer lower prices but I advise against using them. The problem here is that it takes a minimal investment to become a name registrar which can attract minimal investors. A company I once worked for had their email stop working one day because their domain registrar had grown bored with the business and pulled the plug without telling anyone. We never could reach him and it took about a week to get things straightened our and transferred to a new registrar.
The advantage to registering via a registrar separate from your hosting company is that it is a little simpler if you do change hosting companies. If the hosting co is the registrar and you switch you need to transfer your registration from company a to company b which is a little more involved that calling your registrar and telling them "change my host ip from company a to company b"
I don't think it's a big issue and I'm registered with my hosting company.
One thing to keep in mind when selecting a domain name is that they strongly affect search engine rankings. The bad news is that they figured this out years ago so "The Best Deals.com" and "Send me money.com" are already taken. On the other hand if you have a name that no one else will ever use like "Squedle Twaf.com", anytime someone does type it into a search bar you will be right at the top of the results. Of course that does leave the problem of how to get people to type such a word into the search bar, but that is left as an exercise for the student.
Good luck,
Dave Driscoll